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Theorem clmnegsubdi2 25086

Description: Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 29-Sep-2021.)

**Hypotheses**
Ref	Expression
clmpm1dir.v	⊢ 𝑉 = (Base‘𝑊)
clmpm1dir.s	⊢ · = ( ·_𝑠 ‘𝑊)
clmpm1dir.a	⊢ + = (+_g‘𝑊)

**Assertion**
Ref	Expression
clmnegsubdi2	⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = (𝐵 + (-1 · 𝐴)))

**Proof of Theorem clmnegsubdi2**
Step	Hyp	Ref	Expression
1		simp1 1137	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ ℂMod)
2		eqid 2737	. . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2737	. . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
4	2, 3	clmneg1 25063	. . . 4 ⊢ (𝑊 ∈ ℂMod → -1 ∈ (Base‘(Scalar‘𝑊)))
5	4	3ad2ant1 1134	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → -1 ∈ (Base‘(Scalar‘𝑊)))
6		simp2 1138	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉)
7		simpl 482	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ ℂMod)
8	4	adantr 480	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → -1 ∈ (Base‘(Scalar‘𝑊)))
9		simpr 484	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
10		clmpm1dir.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
11		clmpm1dir.s	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑊)
12	10, 2, 11, 3	clmvscl 25069	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ -1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐵) ∈ 𝑉)
13	7, 8, 9, 12	syl3anc 1374	. . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐵) ∈ 𝑉)
14	13	3adant2 1132	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐵) ∈ 𝑉)
15		clmpm1dir.a	. . . 4 ⊢ + = (+_g‘𝑊)
16	10, 2, 11, 3, 15	clmvsdi 25073	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ (-1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉 ∧ (-1 · 𝐵) ∈ 𝑉)) → (-1 · (𝐴 + (-1 · 𝐵))) = ((-1 · 𝐴) + (-1 · (-1 · 𝐵))))
17	1, 5, 6, 14, 16	syl13anc 1375	. 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = ((-1 · 𝐴) + (-1 · (-1 · 𝐵))))
18	10, 11, 15	clmnegneg 25085	. . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → (-1 · (-1 · 𝐵)) = 𝐵)
19	18	3adant2 1132	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (-1 · 𝐵)) = 𝐵)
20	19	oveq2d 7378	. 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((-1 · 𝐴) + (-1 · (-1 · 𝐵))) = ((-1 · 𝐴) + 𝐵))
21		clmabl 25050	. . . 4 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Abel)
22	21	3ad2ant1 1134	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ Abel)
23		simpl 482	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → 𝑊 ∈ ℂMod)
24	4	adantr 480	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → -1 ∈ (Base‘(Scalar‘𝑊)))
25		simpr 484	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
26	10, 2, 11, 3	clmvscl 25069	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ -1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉) → (-1 · 𝐴) ∈ 𝑉)
27	23, 24, 25, 26	syl3anc 1374	. . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (-1 · 𝐴) ∈ 𝑉)
28	27	3adant3 1133	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐴) ∈ 𝑉)
29		simp3 1139	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
30	10, 15	ablcom 19769	. . 3 ⊢ ((𝑊 ∈ Abel ∧ (-1 · 𝐴) ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((-1 · 𝐴) + 𝐵) = (𝐵 + (-1 · 𝐴)))
31	22, 28, 29, 30	syl3anc 1374	. 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((-1 · 𝐴) + 𝐵) = (𝐵 + (-1 · 𝐴)))
32	17, 20, 31	3eqtrd 2776	1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = (𝐵 + (-1 · 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ‘cfv 6494 (class class class)co 7362 1c1 11034 -cneg 11373 Basecbs 17174 +_gcplusg 17215 Scalarcsca 17218 ·_𝑠 cvsca 17219 Abelcabl 19751 ℂModcclm 25043

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-addf 11112 ax-mulf 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-er 8638 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-fz 13457 df-seq 13959 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-0g 17399 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18907 df-minusg 18908 df-mulg 19039 df-subg 19094 df-cmn 19752 df-abl 19753 df-mgp 20117 df-ur 20158 df-ring 20211 df-cring 20212 df-subrg 20542 df-lmod 20852 df-cnfld 21349 df-clm 25044

This theorem is referenced by: (None)

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